Nuclei

Important Topics of Nuclei

Nuclei chapter describes the structure, stability and transformation of the nuclei of atoms as well as the energy emitted in a nuclear process. The major topics are based on nuclear size, mass-energy relation, radioactivity and nuclear reactions, which are very important to Class 12 board exams, JEE Main and NEET. Proper knowledge of these subjects would enable students to understand the basics of nuclear physics and its applications.

1. Composition and Size of the Nucleus

Composition and Size of the Nucleus describes that the nucleus of the atom consists of protons and neutrons, which are jointly referred to as nucleons. It is in this topic that the dependence of the size of a nucleus on its mass number is discussed, as well as the empirical relationship between the nuclear radius and the mass number. It emphasises that the nucleus is very tiny, having a very high density compared to the atom. It is a critical concept on which nuclear structure and stability are taught.

2. Nuclear Density

Nuclear Density is the mass of an atomic nucleus expressed in terms of unit volume. The explanation given in this topic is that all nuclei, regardless of their size or mass number, have almost equal nuclear density. It underlines the extremely large density of the nuclear matter in comparison with the regular substances. The concept of nuclear density can be used to explain how the nuclei are compact.

3. Mass Defect

The difference between the real mass of a nucleus and the mass of the individual protons and neutrons is referred to as Mass Defect. This missing mass is described as energy that is changed into energy during the creation of the nucleus. It gives first-hand evidence of the principle of mass-energy equivalence. Nuclear energy largely depends on mass defect.

4. Binding Energy and Binding Energy per Nucleon

The difference between the real mass of a nucleus and the mass of the individual protons and neutrons is referred to as Mass Defect. This missing mass is described as energy that is changed into energy during the creation of the nucleus. It gives firsthand evidence of the principle of mass-energy equivalence. Nuclear energy largely depends on mthe ass defect.

5. Nuclear Forces

The forces that exist between nucleons (protons and neutrons) within the atomic nucleus are called Nuclear Forces. They are very powerful as compared to electromagnetic forces, but have a very short range of action of the order of nuclear dimensions. It is these forces that keep the nucleus held together, even as the electric repulsive forces exist between the protons. Nuclear forces are weakly dependent on electric charge, as well as exhibit a saturation property.

6. Radioactivity

Radioactivity is the spontaneous and uncontrolled breaking down of unstable nuclei of atoms by releasing α, β, or γ radiations. This reaction is not triggered by any outside factor and is not regulated by temperature, pressure, or chemical condition. Radioactive decay is a random process, and therefore, one cannot know which nucleus will decay at a particular time. In the decay process, the nucleus decays to a more stable nucleus (giving out the energy).

7. Law of Radioactive Decay

The law of radioactive decay says that the radioactive decay rate of a radioactive material at any point is proportional to the quantity of undecayed nuclei of this material at that point in time. This gives a mathematical result of the exponential decay law. The radioactive nuclei gradually decay with time. The decay constant is an individual attribute of the radioactive element and does not vary under any physical condition.

8. Half-Life and Mean Life

Half-life can be defined as the period of time that it takes for the radioactive nuclei in a sample to decay to half of the original amount. Mean life is defined as the average life of a radioactive nucleus till it decays. Both quantities are also independent of any external conditions and only depend on the decay constant. These ideas are most prominently applied to measure the age of radioactive samples as well as in nuclear medicine.

9. Nuclear Reactions

A nuclear reaction is any process where an atomic nucleus is transformed to another nucleus based on the interaction of the nucleus with a particle, i.e. a neutron, proton or a -particle. These reactions include the transformation of mass into energy as per the mass-energy relation of Einstein. Conservation of energy, momentum, electric charge and nucleon number are laws of nuclear reactions. They find application in the fields of research, power generation and medical use.

10. Nuclear Fission

Nuclear fission refers to the breaking up of a heavy nucleus into two or more lighter nuclei as a result of bombarding it with a slow neutron. The energy emitted during this process is huge, in addition to other neutrons, which causes a chain reaction. The emitted energy is a result of the mass difference between the parent nucleus and the products. Nuclear reactors use nuclear fission to generate power and in the manufacture of atomic weapons.

11. Nuclear Fusion

Nuclear fusion involves the coming together of two or more light nuclei to create a heavier nucleus, accompanied by the release of gigantic energy. Very high temperature and pressure are needed in the fusion process to overcome the repulsive electrostatic force between nuclei. The amount of energy emitted is larger than that of fission on the same mass of fuel. Stars obtain their energy through nuclear fusion, which is viewed as a possible source of clean energy in the future.

Related Topics,

Important Formulas of Nuclei

The chapter Nuclei is concerned with the behaviour, structure and properties of atomic nuclei. It describes the ways in which strong nuclear forces hold the protons and neutrons together, and the binding energy determines the stability of the nuclei. This chapter also presents radioactive decay, nuclear reactions, fission, and fusion, which involve massive changes in energy that can be explained by the use of the mass-energy relation by Einstein. Nuclear concepts are the foundation of nuclear energy, radiation technology, as well as numerous uses in medicine and industry.

1. Size of the Nucleus:

Radius of nucleus:

$
R=R_0 A^{1 / 3}
$

where $R_0 \approx 1.2 \times 10^{-15} \mathrm{~m}$

2. Nuclear Density:

$
\rho=\frac{3 m}{4 \pi R^3}
$
Nuclear density is nearly constant for all nuclei.

3. Mass Defect:

$
\Delta m=\left(Z m_p+N m_n\right)-M
$

where
$Z=$ number of protons,
$N=$ number of neutrons,
$M=$ actual mass of nucleus

4. Binding Energy:

$\begin{gathered}\text { Binding Energy (B.E.) }=\Delta m c^2 \\ 1 \mathrm{amu}=931.5 \mathrm{MeV}\end{gathered}$

5. Binding Energy per Nucleon:

B.E. per nucleon $=\frac{\text { Total Binding Energy }}{A}$

6. Law of Radioactive Decay:

$
\frac{d N}{d t}=-\lambda N
$
Integrated form:

$
N=N_0 e^{-\lambda t}
$

7. Activity of a Radioactive Sample:

$
A=\lambda N
$

• Unit: Becquerel (Bq)

8. Half-Life:

$T_{1 / 2}=\frac{\ln 2}{\lambda}=\frac{0.693}{\lambda}$

9. Mean Life:

$
\tau=\frac{1}{\lambda}
$

• Relation:

$
\tau=\frac{T_{1 / 2}}{0.693}
$

10. Relation Between Half-Life and Number of Nuclei:

• After n half-lives:

$N=N_0\left(\frac{1}{2}\right)^n$

11. Energy–Mass Equivalence:

E = mc²

12. Q-value of a Nuclear Reaction:

$
Q=\left(m_{\text {initial }}-m_{\text {final }}\right) c^2
$

• $Q>0$ : Exothermic reaction
• $Q<0$ : Endothermic reaction

Nuclei: Previous Year Questions

Past year questions of the chapter Nuclei are based on basic concepts, including radioactive decay, half-life, binding energy, and nuclear reactions. These questions assist the students to learn the pattern of the exam and finding out the most tested formulas and derivations. Their practice enhances the ability to solve numerical problems and the clarity of concepts. They are important in achieving good scores in board and competitive examinations.

Question 1:

The mean lives of a radioactive substance are 1620 and 405 years for $\beta$-emission and $\beta$-emission, respectively. The time after which three-fourths of a sample will decay if it is decaying both by $\beta$-emission and $\beta$-emission simultaneously will be $\_\_\_\_$ years.

Solution:

$\begin{aligned} & \frac{1}{T}=\frac{1}{T_\alpha}+\frac{1}{T_\beta} \\ & \Rightarrow \mathrm{T}=\frac{T_\alpha T_\beta}{T_\alpha+T_\beta}=324 \text { years } \\ & \frac{N}{N_0}=e^{-\lambda t} \\ & t=\frac{1}{\lambda} \ln \frac{N_0}{N}=T \ln \frac{N_0}{N} \\ & \mathrm{t}=324 \times 2 \ln 2 \\ & \mathrm{t}=449.06 \text { years } \\ & \mathrm{t} \approx 449 \text { years }\end{aligned}$

Question 2:

Find the binding energy per nucleon for ${ }_{50}^{120} \mathrm{Sn}$. Mass of proton $m_p=1.00783 \mathrm{U}$, mass of neutron $m_n=1.00867 U$ and mass of tin nucleus $m_{S n}=119.902199 U$ (take $1 U=931 \mathrm{MeV}$)

Solution:

$\begin{aligned} & \text { B.E. }=[\Delta m] \cdot c^2 \\ & M_{\text {expected }}=Z M_p+(A-Z) M_n \\ & =50[1.00783]+70[1.00867] \\ & M_{\text {actual }}=119.902199 \\ & B . E .=[50[1.00783]+70[1.00867]-119.902199] \times 931 \\ & =1020.56 \\ & \frac{B E}{\text { nucleon }}=\frac{1020.56}{120} \\ & =8.5 \mathrm{MeV}\end{aligned}$

Question 3:

If the binding energy per nucleon in ${ }_3 \mathrm{Li}^7$ and ${ }_2 \mathrm{He}^4$ nuclei are respectively 5.60 MeV and 7.06 MeV, then the energy of the proton in the reaction ${ }_3 \mathrm{Li}+p \rightarrow 2_2 \mathrm{He}^4$ is

Solution:

Applying the principle of energy conservation.
Energy of proton $=$ total BE of $2 \alpha-$ energy of Li

$
\begin{aligned}
& =8 \times 7.06-7 \times 5.6 \\
& =56.48-39.2=17.28 \mathrm{MeV}
\end{aligned}
$

Nuclei in Different Exams

Nuclei chapter is a critical component of Class 12 Physics, and it is crucial in explaining nuclear structure, radioactivity, and nuclear energy. The questions are common in board tests and competitive tests, and are primarily in the form of tests of conceptual clarity, fundamental formulas, and applications. Being aware of the exam-wise concentration and weightage will enable the students to prepare better and score with confidence.

Important Books and Resources for Class 12 Nuclei

To excel in the Nuclei chapter of Class 12 Physics, students should refer to a balanced set of NCERT textbooks, reference books, and practice materials that explain both basic concepts and their applications. These books include major concepts of nuclear structure, nuclear forces, radioactivity, decay laws, binding energy, and nuclear reactions that are needed in Class 12 board tests, JEE Main, JEE Advanced and NEET.

NCERT Resources for Nuclei

NCERT resources for the Nuclei chapter are the most authoritative and syllabus-aligned study materials for understanding nuclear structure, radioactivity, binding energy, decay laws, and nuclear reactions as prescribed in the Class 12 Physics curriculum. These topics are well explained and demonstrated using diagrams and examples directly as outlined in the syllabus in the NCERT textbook and exemplar problems. One should prepare extensively using NCERT books in order to score highly in Class 12 Board examinations and competitive exams such as JEE Main and NEET.

• NCERT solutions for Class 12 Physics Chapter 13 Nuclei
• NCERT notes Class 12 Physics Chapter 13 Nuclei
• NCERT Exemplar Class 12 Physics Solutions Chapter 13

NCERT Subjectwise Resources

NCERT subject-wise materials are organised and syllabus-based learning content on various subjects, which assists students in developing a good conceptual basis. They consist of textbooks, exemplar problems, and solutions and can thus be very helpful in the preparation for the board exams and even competitive exams such as JEE and NEET.

Practice Questions based on Nuclei

Practice Questions on the Nuclei Chapter help students strengthen their understanding of nuclear structure, stability, and energy-related processes. These questions mainly focus on mass defect, binding energy, radioactive decay law, half-life, nuclear reactions, and fission-fusion concepts. Regular practice improves conceptual clarity, numerical accuracy, and confidence in applying formulas. The solution of different types of problems is very useful not only for the Class 12 board examination but also for competitive examinations such as JEE Main and NEET.

Conclusion

The chapter Nuclei develops a solid conceptual background of the way atoms are organised to form nuclei and the energy used during nuclear processes. Students can gain clear and precise conceptual knowledge by updating important formulas, fundamental ideas, and theoretical explanations associated with nuclear size, mass defect, binding energy, radioactivity, and nuclear reactions. This focused preparation enhances confidence and proves highly effective for performing well in Class 12 board examinations as well as competitive exams like JEE Main and NEET.